Title: Optima
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Permanent Link: http://ufdc.ufl.edu/UF00090046/00019
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Title: Optima
Series Title: Optima
Physical Description: Serial
Language: English
Creator: Mathematical Programming Society, University of Florida
Publisher: Mathematical Programming Society, University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: September 1986
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Bibliographic ID: UF00090046
Volume ID: VID00019
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Holding Location: University of Florida
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Are you all salesmen, here?

The Book is Not Closed
on the TSP
E.L. Lawler
University of California,
J.K. Lenstra
Centre for Mathematics &
Computer Science, Amsterdam
A.H.G. Rinnooy Kan
Erasmus University, Rotterdam
D.B. Shmoys
Massachusetts Institute of
Technology, Cambridge

N view of the extensive literature
devoted to The Traveling
Salesman Problem [1] over the
last thirty years, with roots as
early as 1856, one would expect
that all of the most interesting
questions concerning the TSP had
been settled long ago. Nothing
could be further from the truth. In
Sherlockian terms, the recently published
textbook [1] is no more than A Study in
Scarlet, and even when The Final Problem
has been solved, one can await The Return
of the Traveling Salesman.
The book is certainly not closed on the
TSP. We offer the following collection of
problems as a guide to further work on
combinatorial optimization's most
notorious problem. The collection
parallels our reference [1] in structure.
For the uninitiated, the TSP is the
problem of finding the shortest tour
visiting each of n cities exactly once, when
the intercity distances are given. We will
denote the distance from city i to city j by
cij. The cost of following the route
corresponding to the cyclic permutationT
will be denoted by c(z) = l=1 ci, (i).

One of the mysteries surrounding the
TSP is its origin, at least in its most recent
incarnation. Current thought traces the
path back to Hassler Whitney, but unless
Whitney's (unpublished) notes provide any
definite confirmation, it appears to be a
difficult task to prove this conjecture.
For the bottleneck TSP, the aim is to
find a tour through all cities that minimizes
the longest intercity distance between two
consecutive visits. It is known how to
model the bottleneck TSP as a single
ordinary TSP with the same number of
cities using distances whose length (i.e.
number of binary digits) is polynomially
bounded. However, the converse is not
known: How can one model the ordinary
TSP as a bottleneck TSP using even
polynomially many cities and distances of
polynomial length? As a greater challenge,
can either of these transformations be done
such that the distances themselves are
polynomially bounded?
(It turns out that the first of these two
questions is challenging enough. A
remarkable recent result of Krentel (to
appear in the Proceedings of the 18th
Symposium on the Theory of Computing)
implies that the existence of any
polynomial method to model the ordinary
TSP as a bottleneck TSP would imply that
P = NP. This negative result, for all intents
and purposes, closes this problem.)
One popular form of the TSP is the
Euclidean TSP, where each city i is
specified by a location (xi,Yi) in Euclidean
space with rational coordinates, so that
cj = {(xi x)2 + (Yi yj)2). It is
unknown whether the TSP in this form is
in NP. The fundamental problem here is to
find a polynomial procedure to decide
whether c(z) rational k that are input to the procedure.
Circulant matrices (cij) can be
characterized in the following way: if the
continued on page two

By now most of the combinatorial
optimization community is aware that E.R.
Swart of the University of Guelph has
produced a paper asserting a proof that the
famous complexity classes P and NP are
equal. Given a graph on n vertices, Swart
presents a system of linear constraints over
nonnegative variables which he argues has
a solution if and only if the given graph is
Hamiltonian. Since this system has a
polynomial (in n) number of constraints
and variables, the paper implies the
classically NP-Complete question of
whether a graph is Hamiltonian can be
polynomially resolved by direct application
of available (Khachian and Karmarkar)
linear programming algorithms.
If correct, Swart's results could be
conservatively described as startling and
revolutionary. An array of fundamental
issues in discrete mathematics would be
resolved in a single advance. It is only
professional to be suspicious, but (at this
writing) we know of no one who has either
verified Swart's approach or proved it
fatally flawed.
As part of the recent AFOSR sponsored
Advanced Research Institute on Discrete
Applied Mathematics (ARIDAM) at the
Rutgers Center for Operations Research
(RUTCOR) seven of us: (D. Crystal, H.
Greenberg, A. Kolen, W. Morris, A.
Rajan, R. Rardin, M. Trick) did have an
opportunity to discuss the paper directly
with Swart and study it carefully amongst
ourselves. Our summary of the paper,
entitled "On the P=NP Paper by E.R.
Swart," is available by writing to Ronald
L. Rardin, School of Industrial
Engineering, Purdue Univ., West Lafayette,
IN 47907. Copies of Swart's paper may be
obtained by writing to him at the
Department of Mathematics and Statistics,
Univ. of Guelph, Guelph, Ontario, Canada.
-R. Rardin

_ ~~ ~~

Are you all salesmen, here?from page one
Are you all salesmen, here? from page one

first row of the matrix has entries ao,
al...... an.l, then all of the entries cij
such that j i = k mod n are equal to ak.
For intercity distances given by circulant
matrices, the nearest neighbor rule finds a
shortest path visiting each city exactly
once and thereby solves what is
sometimes called the Wandering Salesman
Problem. It is unknown, however,
whether there is a polynomial procedure to
find a shortest tour.

A NOTHER natural constraint on the
intercity distances is the triangle inequality;
that is, cik < ciJ + cjk for all ij,k. An
annoyingly simple (but open) question is
whether there exists a polynomial
approximation algorithm which is
guaranteed to find a tour t such that c(r) <
kc(T*) where T* is an optimal tour and k
is a constant.
A classic result of Beardwood, Halton
and Hammersley states that, for n points
distributed independently and uniformly
over the unit square, the Euclidean length
of the shortest tour through these points is,
with probability 1, asymptotically equal to
Pqn for some constant p. However, the
proof is existential and does not provide the
value of p. Although rough bounds are
known for p, the precise value remains
elusive and finding it is almost surely a
formidable challenge.
An often used approach to find good
approximate tours involves neighborhood
search, where the current tour is tested for
improvement by exchanging a handful of
arcs. Perhaps the most successful
procedure of this type is due to Lin and
Kemighan. In order to find such a locally
optimal solution, it appears to be necessary
to repeatedly apply the test for
improvement, generate the next solution,
and then proceed iteratively. Does there
exist a fast algorithm which directly
generates a local optimum?

TLHE principal aim of the polyhedral
approach to the TSP is the identification of
facets of the convex hull of incidence
vectors of tours. These tours are
Hamiltonian cycles in a graph with the
cities as vertices and the intercity distances
as arc weights. Complexity theory
suggests that, in the case of a complete
graph, a complete characterization of all the
facets is a hopeless task. For special
classes of graphs, however, such
characterizations have been obtained. A
challenging type of research question is to
determine the class of all graphs for which
a given system of 'nice' facets provides a
complete characterization.
A class of rather sophisticated nice
facets is the one defined by clique tree
inequalities. Their algorithmic applica-
tion is obstructed by the fact that the
separation problem has not yet been solved:
Given a vector, check whether it satisfies
all clique tree inequalities; if not, find one
which is violated. A more modest but still
unfulfilled aim would be to find a good
heuristic for this problem.

A NY optimization algorithm for the
TSP is likely to require some form of
enumeration of the solution set. Most
enumerative methods employ the branch
and bound principle. The main ingredient
of such a method is a relaxation of the
TSP, whose solution yields a lower bound
on the length of the shortest tour. In the
case of the asymmetric TSP with
independently distributed intercity distances,
all successful branch and bound algorithms
use the relaxation to the linear assignment
problem. In the symmetric case, where c,
= c; for all ij, the strongest bounds are
obtained from the 1-tree relaxation with
Lagrangean objective or from the
continuous 2-matching relaxation with the
addition of facet defining inequalities. So
far, no one has succeeded in developing a
robust branch and bound algorithm, which
performs well in both cases.
A special case of the TSP that was
considered before the TSP itself is the
problem of finding a Hamiltonian cycle in
a graph. A graph is called t-tough if

deleting any set of s vertices leaves the
graph with at most sit connected
components. Does there exist some t such
that all t-tough graphs are Hamiltonian? In
particular, are all 2-tough graphs

STSP finds application in the
context of vehicle routing, which is a broad
and diverse area. We offer a theoretical and
a practical challenge. First, consider the
Chinese multipostman problem. Given an
edge-weighted connected graph, determine a
set of m tours, each starting and finishing
at a specified vertex, collectively traversing
all edges, and no one exceeding a given
limit in length. The problem is NP-hard in
the strong sense when m is arbitrary. What
is its complexity status when m is fixed?
Secondly, vehicle routing often involves
the problem of period allocation. Each
customer requires a given number of visits
in a given period. On which day should
each customer be visited such that the
collection of solutions to the daily routing
problems is overall optimal? This is a
neglected problem, and finding good
heuristics for its solution is of eminent
practical importance.
The problems given above represent
just a sample of the work to be done on the
TSP. One problem that will probably
never be solved is that along with its eye-
grabbing name, the traveling salesman
problem will never rid itself of associated
off-color jokes and inevitable
misunderstanding. No waitress would ever
inquire of a group of editors who are
casting a first glance at their book, The
Minimum Spanning Tree Problem, 'Are
you all trees, here?'

1. EL. Lawler, J.K. Lenstra,
A.H.G. Rinnooy Kan, D.B. Shmoys
(eds.) (1985).
The Traveling Salesman Problem:
A Guided Tour of Combinatorial
Wiley, Chichester.

Confeirencel Notes
__________________~"~`"""~~~; '~"' '"`""~` "" ~"~'"=~ ; `-'- ';-" I'

Call For Papers
11th Triennial
Conference on
Operations Research
Buenos Aires, Argentina
August 10-14, 1987
The International Federation
of Operational Research
Societies (IFORS) will be 28
years old in 1987. As an
association of 34 national OR
societies and 6 kindred societies,
its purpose is the development
of operations research as a
unified science and its
advancement in all nations of
the world. One of IFORS'
main activities is the orga-
nization of an international
conference every three years.
The last conference was in
Washington, D.C. You are now

invited to the next one to be
held in Buenos Aires.
Papers may be contributed
by any member of an OR
society affiliated with IFORS.
Authors are requested to submit
an abstract of not more than
100 words, with 5 key words,
not later than October 13, 1986,
directly to the Chairman of the
Program Committee, M.E.
Thomas, ISyE, Georgia Tech,
Atlanta, GA 30332. The dead-
line will be strictly enforced.
The Program Committee re-
serves the right to accept or
reject these contributions on the
basis of the abstract or paper.
All the abstracts of the papers
to be presented will be
published in advance of the
conference if they contain no
more than 100 words.

Martin Beale Memorial

A symposium will be held
at The Royal Society, London,
July 6-8, 1987, in memory of
Professor E.M.L. Beale. By
covering the range of his
professional interests, the
symposium will provide useful
links between research and
applications in Mathematical
Programming, Operational
Research and Statistics. The
programme will include plenary
talks for a general audience,
submitted papers in parallel
sessions, and a conference
dinner. For further information
please contact Mrs. B.A.
Peberdy, SCICON Limited,
Wavendon Tower, Milton
Keynes MK17 8LX, U.K.
-M.J.D. Powell


Alan Hoffman
Honored by

Dr. Alan Hoffman from
IBM Watson Research Center at
Yorktown Heights was awarded
an Honorary Doctorate of
Science by the Technion Israel
Institute of Technology. The
ceremony took place at
Technion City in Haifa, Israel,
on June 16, 1986. The citation
for the award reads, "In
recognition of his fundamental
contributions to Operations
Research and its interface with
Combinatorics, Graph Theory
and Linear Algebra; in
particular, for the introduction
of unimodularity, thereby
laying the foundation for Integer
-Uriel Rothblum

Journals and Studies

Volume 35, No. 2

S.E. Berenguer and R.L. Smith, "The Expected Number of
Extreme Points of a Random Linear Program."
C. Blair, "Random Inequality Constraint Systems with few
N. Megiddo, "Improved Asymptotic Analysis of the Average
Number of Steps Performed by the Self-Dual Simplex Algorithm."
MJ. Todd, "Polynomial Expected Behavior of a Pivoting
Algorithm for Linear Complementarity and Linear Programming
C.A. Tovey, "Low Order Polynomial Bounds on the Expected
Performance of Local Improvement Algorithms."
N. Megiddo, "On the Expected Number of Linear
Complementarity Cones Intersected by Random and Semi-random
Y-H. Wan, "On the Average Speed ofLemke's Algorithm for
Quadratic Programming."

Volume 35, No. 3

M. Fukushima, "A Successive Quadratic Programming Algorithm
with Global and Superlinear Convergence Properties."
MJ.D. Powell and Y. Yuan, "A Recursive Quadratic
Programming Algorithm that Uses Differentiable Exact Penalty
S. Sen and H.D. Sherali, "A Class of Convergent Primal-dual
Subgradient Algorithms for Decomposable Convex Programs."
S. Fujishige, "A Capacity-Rounding Algorithm for the Minimum-
Cost Circulation Problem: A Dual Framework of the Tardos

A. Ruszczynski "A Regularized Decomposition Method for
Minimizing a Sum of Polyhedral Functions."
R. Lazimy, "Solving Multiple Criteria Problems by Interface
M.L. Balinski, Th. M. Liebling, and A.-E. Nobs, "On the
Average Length of Lexicographic Paths."
N. Megiddo, "A Note on Degeneracy in Linear Programming."
S. Zlobec, "Characterizing an Optimal Input in Perturbed
Convex Programming: Corrigendum."

Volume 36, No. 1

G. di Pillo, "An Exact Penalty Function Method with Global
Convergence Properties for Nonlinear Programming Problems."
J. Herskovits, "A Two-Stage Feasible Directions Algorithm for
Nonlinear Constrained Optimization."
C. Brezovec, G. Coruejols, F. Glover, 'Two Algorithms for
Weighted Matroid Intersection."
J-S. Pang, "Inexact Newton Methods for the Nonlinear
Complementarity Problem."
A. Dax, "A Note on Optimality Conditions for the Euclidean
Multifacility Location Problem."
O.L. Mangasarian and T.-H Shiau, "Error Bounds for Monotone
Linear Complementarity Problems."
J.-P. Dussault, J.A. Ferland, B. Lemaire, "Convex Quadratic
Programming with one Constraint and Bounded Variables."
J. Kyparisis, "Uniqueness and Differentiability of Solutions of
Parametric Nonlinear Complementarity Problems."
H.A. Eiselt, "Continuous Maximum Knapsack Problems with
GLB Constraints."

~ ---

Technical Reports and Working Papers M

Stichting Mathematisch Centrum
Centrum voor Wiskunde en Informatica
Department of Operations Research and
System Theory
Postbus 4079 1009 AB Amsterdam
Kruislaan 413 1098 SJ Amsterdam

G.A.P. Kindervater and J.K. Lenstra, "An Introduction to
Parallelism in Combinatorial Optimization," OS-R8501.
J.M. Schumacher, "A Geometric Approach to the Singular
Filtering Problem," OS-R8502.
B.J. Lageweg, J.K. Lenstra, A.H.G. Rinnooy Kan e.a.,
"Stochastic Integer Programming by Dynamic Programming," OS-
J.B. Orlin, "Genuinely Polynomial Simplex and Non-Simplex
Algorithms for the Minimum Cost Flow Problem," OS-R8504.
J.B. Orlin, 'The Complexity of Dynamic/Periodic Languages and
Optimization Problems," OS-R8505.
J.M. Schumacher, "Residue Formulas for Meromorphic Matrices,"
J.H. van Schuppen, "Stochastic Realization Problems Motivated
by Econometric Modelling," OS-R8507.
R.K. Boel and J.H. van Schuppen, "Overload Control for SPC
Telephone Exchanges Refined Models and Stochastic Control," OS-
J.W. Polderman, "A Note on the Structure of Two Subsets of the
Parameter Space in Adaptive Control Problems," OS-R8509.
O.J. Boxma and B. Meister, "Waiting-time Approximations for
Cyclic-Service Systems with Switch-Over Times," OS-R8510.
O.J. Boxma, "A Queueing Model of Finite and Infinite Source
Interaction," OS-R8511.
G.A.P. Kindervater and H.WJ.M. Trienekens, "Experiments with
Parallel Algorithms for Combinatorial Problems," OS-R8512.
P.J.C. Spreij, "Recursive Parameter Estimation for Counting
Processes with Linear Intensity," OS-R8513.

Georgia Institute of Technology
School of Industrial and Systems Engineering (ISyE)
College of Management (COM)
School of Mathematics (Math)
School' of Information and Computer Science (ICS)

F.A. Al-Khayyal, TJ. Hodgson, G.D. Capps, J.A. Dorsch, D.A.
Kriegman and P.D. Pavnica, "A Lagrangian Dual Approach for
Solving a Structured Geometric Program Arising in Sample Survey
Design," ISYE Report Series No. J-85-1.
F.A. Al-Khayyal, "Necessary and Sufficient Conditions for the
Existence of Complementary Solutions and Characterizations of the
Matrix Classes Q and QO," PDRC Report Series 85-5, ISyE.
F.A. Al-Khayyal, "Minimizing a Quasiconcave Function Over a
Convex Set: A Case Solvable by Lagrangian Duality," ISyE.
F.A. Al-Khayyal, C.A. Tovey, "Extending Simulated Annealing
to Continuous Global Optimization," ISyE.
M.A. Berger and A. Felzenbaum, "Sign Patterns of Matrices and
Their Inverses," Math.
M.A. Berger, A. Felzenbaum and A.S. Fraenkel, "Lattice

Parallelepipeds and Disjoint Covering Systems," Math.
M.A. Berger, A. Felzenbaum and A.S. Fraenkel, "A Non-analytic
Proof of the Newman-Znam Result for Disjoint Covering Systems,"
M.A. Berger, A. Felzenbaum and A.S. Fraenkel, "Improvements
to the Newman-Znam Result for Disjoint Covering Systems," Math.
M.A. Berger, A. Felzenbaum and A.S. Fraenkel, "Covers of
Product Sets and the Korec-Znam Result," Math.
M.A. Berger, A. Felzenbaum and A.S. Fraenkel, "Disjoint
Covering Systems with Precisely One Multiple Modulus," Math.
M.A. Berger, "A Trotter Product Formula for Random Matrices,"
C.E. Blair, R.G. Jeroslow, and J.K. Lowe, "Some Results and
Experiments in Programming Techniques for Propositional Logic,"
University of Illinois and COM, Georgia Inst. Tech.
C.Cardelino and P.Y. Chen, "Algorithm for Solving a Complex
Function F(z)=O," IT-ICS-85/04.
D. Crystal, "Min Algebraic Duality," ISyE.
M. Graham, N. Griffeth, and B. Smith-Thomas, "Algorithms for
Providing Reliability in Unreliable Database Systems," IT-ICS-
W. Green and E.W. Kamen, "Stabilizability of Linear Systems
Over a Commutative Normal Algebra with Applications to Spatially-
Distributed and Parameter-Dependent Systems," Math, Georgia
Institute of Technology and Electrical Engineering, University of
W. Green and T.D. Morley, "Operator Means, Fixed Points and
the Norm Conveyance of Monotone Approximants," Math.
N. Griffeth and J.A. Miller, "Performance Modeling of Database
Recovery Protocols," IT-ICS-84/21.
R. Jeroslow, "Computation-oriented Reductions of Predicate to
Propositional Logic," COM.
R. Jeroslow, "Alternative Formations of Mixed Integer
Programs," COM.
R. Jeroslow, "A Simplification for some Disjunctive
Formulations," COM.
R. Jeroslow, "Spatial Embeddings for Linear and Logic
Structures," COM.
R. Jeroslow, "On Monotone Chaining Procedures of the CF
Type," COM.
R. Jeroslow, "An Extension of Mixed-Integer Programming
Models and Techniques to Some Database and Artificial Intelligence
Settings," COM.
J. Miller and N.G. Griffeth, "Approximate Queuing Network
Analysis of Database Recovery Protocols," IT-ICS-85/12.
J. Miller and N. Griffeth, "Markovian Analysis and
Optimization of Database Recovery Protocols," GIT/ICS/86/0Z.
D.C. Nachman, "Arbitrage Operations and Market Expansion,"
D.C. Nachman, "Options and Market Expansion," COM.
D.C. Nachman, "Efficient Funds for Meager Asset Spaces,"
M.B. Richey and R.G. Parker, "On Multiple Steiner Subgraph
Problems," ISyE Report Series.
M.B. Richey and R.G. Parker, "Minimum-Maximal Matching on
Series-Parallel Graphs: An Algorithm and Its Implications for Other
Problems," ISyE Report Series.
M.B. Richey and R.G. Parker, "Solving General Eulerian
Subgraph Problems," ISyE Report Series.
continued on following pages

~I I ~ ~ I---

J.E. Spingam and J. Lawrence, "On Fixed Points of
Nonexpansive Piecewise Isometric Mappings," Math., Georgia Tech,
and Dept. of Mathematics, George Mason University.
G. ViJi.. .in. "Some Properties and Conjectures of the Geometry
of Planar Graphs," IT-ICS-85/17.

University of Bonn
Department of Operations Research
Nassestr. 2
D-5300 Bonn 1, West Germany

A.M.H. Gerards and A. Schrijver, "Matrices with the Edmonds-
Johnson Property," WP 85363.
B. Korte and L. Lovasz, "Homomorphisms and Ramsey
Properties of Antimatroids," WP 85364.
K. Vesztergombi, "Bounds on the Number of Small Distances in
a Finite Planar Set," WP 85365.
U. Derigs and A. Metz, "On the Use of Optimal Fractional
Matchings for Solving the (Integer) Matching Problem,"
WP 85366.
U. Derigs and A. Metz, "An Efficient Labeling Technique for
Solving Sparse Assignment Problems," WP 85367.
L. Lovasz, "An Algorithmic Theory of Numbers, Graphs, and
Convexity," WP 85368.
W. Cook, A.M.H. Gerards, A. Schrijver and E. Tardos,
"Sensitivity Results in Integer Linear Programming," WP 85369.
B. Korte, "Was ist Kombinatorische Optimierung," WP 85370.
F. Barahona and A.R. Mahjoub, "Compositions in the Acyclic
Subdigraph Polytope," WP 85371.
U. Faigle, "Exchange Properties of Combinatorial Closure
Spaces," WP 85372.
U. Faigle and Gy. Turn, "On the Complexity of Interval Orders
and Semiorders," WP 85373.
A. Frank and E. Tardos, "An Application of the Simultaneous
Approximation in Combinatorial Optimization," WP 85375.
U. Derigs and A. Metz, "An In-Core/Out-of-Core Method for
Solving Large Scale Assignment Problems," WP 853676.
W. Cook, C.R. Coullard and Gy. Turn, "On the Complexity of
Cutting-Plane Proofs," WP 85377.
U. Faigle, "Matroids in Combinatorial Optimization,"
WP 85378.
D. Crystal and L.E. Trotter, Jr., "On Abstract Integral
Dependence," WP 85379.
U. Faigle and R. Schrader, "Setup Minimization Techniques for
Comparability Graphs," WP 85380.
M.M. Syslo, "Remarks on Dilworth Partially Ordered Sets," WP
U. Faigle and R. Schrader, "Interval Orders without Odd Crowns
are Defect Optimal," WP 85382.
A. Sebi, "A Very Short Proof of Seymour's Theorem on
t-joins," WP 85383
B. Korte and L. Lavasz, "On Submodularity in Greedoids and a
Counterexample," WP 85384.
E. Tardos, C.A. Tovey, and M.A. Trick, "Layered Augmenting
Path Algorithms," WP 85385.
U. Faigle and R. Schrader, "On the Computational Complexity of
the Order Polynomial," WP 85386.

W. Schmidt, "A Characterization of Undirected Branching
Greedoids," WP 85387.
W. Schmidt, "Greedoids and Searches in Directed Graphs,"
WP 85388.
B. Korte and L. Lovisz, "Polyhedral Results for Antimatroids,"
WP 85390.
U. Faigle, "Submodular Combinatorial Structures," WP 85392.
U. Faigle and R. Schrader, "Greene-Kleitman Extensions of
Ordered Sets," WP 85393.
A. Seba, "The Schrijver System of Odd Join Polyhedra,"
WP 85394.
A.M.H. Gerards and A. Seb6: "Total Dual Integrality Implies
Local Strong Unimodularity," WP 85395.
W. Schmidt, "A Min-Max Theorem for Greedoids," WP 85396.
Ph. Mahey, "Subgradient Techniques and Combinatorial
Optimization," WP 85397.
H.J Promel, "Some Remarks on Natural Orders for Combinatorial
Cubes," WP 85398.
U. Faigle and R. Schrader, "Setup Optimization Problems with
Matroid Structure," WP 85399.
U. Derigs, "Neuere Ansiitze in der Linearen Optimierung,"
WP 85402.
H.J Pr6mel and B. Voigt, "Graham-Rothschild Parameter Sets,"
WP 85403.
J. Edmonds and A. Lubiw, "Bipartition Systems and How to
Partition Polygons," WP 85406.

Systems Optimization Laboratory
Department of Operations Research
Stanford University
Stanford, California 94305

H. Hu, "Existence of Equilibrium Prices for a Simple Planning
Model," SOL 85-10.
P.E. Gill, W. Murray, M.A. Saunders, J.A. Tomlin and M.H.
Wright, "On Projected Newton Barrier Methods for Linear
Programming and An Equivalence to Karmarkar's Projective
Method," SOL 85-11.
P.H. McAllister, J.C. Stone, G.B. Dantzig and B.Avi-Itzhak,
"Changes Made for the Pilot-1983 Model," SOL 85-12.
B.C. Eaves and U.G. Rothblum, "A Theory on Extending
Algorithms for Parametric Problems," SOL 85-13.
P.E. Gill, S.J. Hammarling, W.Murray, M.A. Saunders and M.H.
Wright, "User's Guide for LSSOL (Version 1.0): A FORTRAN
Package for Constrained Linear Least-Squares and Convex Quadratic
Programming," SOL 86-1.
P.E. Gill, W. Murray, M.A. Saunders and M.H. Wright, "User's
Guide for NPSOL (Version 4.0): A FORTRAN Package for Nonlinear
Programming," SOL 86-2.
M.Aganaic' and R.W. Cottle, "A Constructive Characterization
of Q0-Matrices with Nonnegative Principal Minors," SOL 86-3.
M.S. Bellovin, "Reliability and Shortage Distribution
Computations in General Stochastic Transportation Networks," SOL
Y.-H. Wan, "An Implicit Enumeration Algorithm with Binary-
Valued Constraints," SOL 86-5.


Technical Reports and Working Papers

P.E. Gill, W. Murray, M.A. Saunders and M.H. Wright, "Some
Theoretical Properties of an Augmented Lagrangian Merit Function,"
SOL 86-6.
P.E. Gill, W.Murray, M.A. Saunders and M.H. Wright, "A Note
on Nonlinear Approaches to Linear Programming," SOL 86-7.
P.E. Gill, W. Murray, M.A. Saunders and M.H. Wright,
"Maintaining LU Factors of a General Sparse Matrix," SOL 86-8.
S.C. Hoyle, "A Single-Phase Method for Quadratic
Programming," SOL 86-9.
H. Hu, "An Algorithm for Rescaling a Matrix Positive Definite,"
SOL 86-10.
G.B. Dantzig, "Need to Do Planning Under Uncertainty and the
Possibility of Using Parallel Processors for this Purpose,"
SOL 86-11.
H. Hu, "An Algorithm for Positive Definite Least Square
Estimation of Parameters," SOL 86-12.

Northwestern University
Department of Industrial Engineering
and Management Sciences
Evanston, Illinois 60201

W.J. Hopp, "Sensitivity Analysis in Discrete Dynamic
Programming," TR 85-07, November 1985.
P.C. Jones, J.L. Zydiak and WJ. Hopp, "Prices and
Depreciation for Machine Replacement I: Deterministic Case,"
TR 85-08, December 1985.
F. Fourer, "A Simplex Algorithm for Piecewise-Linear
Programming III: Computational Analysis and Applications,"
TR 86-03, June 1986.
W.-L. Hsu, "Recognition and Optimization Algorithms on
Planar Perfect Graphs," TR 86-04, June 1986.
R.E. Bixby and R. Fourer, "Finding Embedded Network Rows
in Linear Programs I: Extraction Heuristics," TR 86-05, July 1986.

Book Ri:.i

Combinatorial Optimization for Undergraduates
By L.R. Foulds
Undergraduate Texts in Mathematics
Springer, Berlin, 1984
ISBN 0-387-90977-X

This text covers a broad range of topics in combinatorial
optimization as may be inferred from the following chapter titles:
0. Introduction; 1. Linear Programming and Extensions (including
the transportation and assignment problems); 2. Solution
Techniques (integer and dynamic programming, complexity,
heuristics); 3. Optimization on Graphs and Networks (trees, paths,
flows, CPM and PERT); 4. Some Applications (facilities layout,
traveling salesman, vehicle routing, capacitated trees, evolutionary
trees); 5. Appendix (linear algebra and graph theory).
There are some positive aspects of this effort by Foulds. We
agree with him that there is a need for a good undergraduate level
text devoted to combinatorial optimization. We also admire his
attempt to motivate theory through small examples. Nevertheless,
we feel that the book does a disservice to the area that it purports to
serve. It suffers from a number of major deficiencies.
The first paragraph of the text sets the stage. It paraphrases a
continuous optimization problem from Virgil's Aeneid that is
solvable by variational calculus a truly surprising motivation for
the study of combinatorial optimization. The remainder of the
introductory chapter will most likely confuse any novice in the area
of optimization. After a notationally laborious description of the
shortest Hamiltonian path problem, Foulds provides some
techniques for local improvement. He then gives the following
definition: "If, for all such xi f(xo)< f(xi), xo is said to be a local
minimum of f." But there is no antecedent for the word "such".

Indeed, one would have to know the meaning of the term "local
minimum" to find sense in the definition.
Chapter 1 falls short of any expectations raised by its title.
The treatment of linear programming lacks care at the level of
detail. It is confusing to read "b > 0" where "b > 0" is intended
(page 25). It is troublesome to work with lower as well as upper
case notation for vectors (but, if it has to be done, then using both
for one and the same vector is a good exercise (page 36)). It is
frustrating to try to apply phase 2 of the simplex method without
being told first how artificial variables should be removed from the
basis (page 29). And it is impossible to solve assignment
problems to optimality by a quasi-Hungarian method that relies on a
simple greedy heuristic for finding minimum bipartite vertex covers
(pages 73, 74). Proclaiming optimality of this rule is more than a
detail and worse than lack of care.
We are also displeased with the section on complexity in
Chapter 2. At the technical level, the definition of P is wrong, that
of NP is lacking, and the statement, "If pl is NP-complete and pl cc
P2 then P2 is also NP-complete," is false. Our main objection,
however, is that by making one-and-a-half pages of inexact
statements about complexity and by not integrating the concept
into the rest of the text, Foulds has mystified rather than clarified
the subject.
The next section, on heuristics, is not satisfactory either. The
main message here is that one has to be realistic and clever when
solving problems in business and industry. While we do not
disagree with realism or creativity, we do disagree with the
looseness of the approach. No attempt is made to provide a
theoretical framework of heuristics, and no word is spent on worst
case or probabilistic analysis. All this is saddening. A formal
treatment of computational complexity and approximation

I ~ ~ ~

r k I

.IIL .rilimi can be so easy, so illuminating, and so exciting.
Chapter 5 contains an eclectic collection of toy applications
that are solved by straightforward heuristics, mostly of a greedy
nature. The presentation falls short on two counts: the techniques
introduced in the previous chapters are largely disregarded, and the
reader is not even informed about the existence of more
sophisticated solution methods.
We strongly recommend against the adoption of this text for
any course in combinatorial optimization.
-J.K. Lenstra, J.B. Orlin

Linear and Nonlinear Programming
By R. Hartley
Horwood, Chichester, 1985
ISBN 0-85312-644-5

This nice little book gives an informal introduction to linear
programming and its main related topics. The presentation is done
with a minimum of mathematical prerequisites without any formal
statement and proof of theorem; therefore, as such, the book may be
attractive to lecturers having to face students majoring in business
administration, economics or engineering where mathematical
formalism is not always much appreciated. The book is written in a
clear and easily readable manner. Each chapter is followed by
exercises for which hints and answers can be found at the end of the
book. With its 221 pages, the book covers quite a lot of material.
After an introduction where linear programming is presented via
a "real world" example and is illustrated graphically, the first four
chapters are devoted to the simplex method including the revised
version. All this is done without using matrix notation. Chapters
5 to 7 are concerned with duality, sensitivity analysis and bounded
variables. Related standard topics such as the transportation
problem, the multiobjective and the integer programming cases are
treated in chapters 8 to 11. The last chapter is on quadratic
programming. To have written more about nonlinear programming,
even in an informal manner, probably would have made the book
too long.
-Jacques Gauvin

Tree Automata
By F. Gecseg and M. Steinby
Akademiai Kiad6, Budapest, 1984

Tree automata were invented in the mid-sixties as a natural
generalization of finite automata, accepting finite (valued) trees
instead of strings. Although the resulting theory of recognizable
(or "regular") sets of trees soon offered appealing results and
interesting problems and also provided useful insights into
questions of classical formal language theory, until now it was not
covered by a textbook collecting the relevant notions and most
important theorems.
The book by Gdcseg and Steinby fills this gap. There are four
chapters, each of them supplemented by historical notes and

references to further literature. The first chapter presents some
universal algebra as the terminological background upon which the
exposition of tree automata theory is based. In the second chapter
one finds an introduction to the theory of recognizable forests (sets
of trees), including a comparison of root-to-frontier and frontier-to-
root tree automata, Kleene's theorem in the context of trees,
algebraic characterizations of recognizability, and several theorems
concerning minimal tree automata. The following chapter deals
with applications of these results to context-free (string-)
languages. In the fourth chapter, tree transducers are investigated.
First, Engelfriet's classification of tree transformations and the
fundamental decomposition results for these relations are presented.
Then follows a discussion of tree transducers with regular look-
ahead, surface forests, hierarchies of tree transformations, and the
equivalence problem for tree transducers. The book ends with an
extensive bibliography consisting of about 280 items which covers
virtually all relevant literature up to 1982.
It should be noted that some major developments of the subject
which are important in current research are only mentioned in the
notes but not treated in the main text. In particular, this applies to
automata on infinite trees, the connection with logic monadicc
second-order logic and logics of programs), and issues related to the
concept of context-free tree grammar, like pushdown tree automata
or macro tree transducers.
Nevertheless, this carefully written monograph contains all that
could be called the "basic theory" of tree automata; it will be a very
useful reference for anyone who is interested in this rapidly
developing area.
-W. Thomas

Introduction to Stochastic Dynamic Programming
By Sheldon M. Ross
Academic Press, London, 1983

This book provides an introduction to the theory of stochastic
dynamic programming or Markov decision processes. As can be
seen from the table of contents, both classical and modem fields of
research are covered as follows: I. Finite-Stage-Models; II.
Discounted Dynamic Programming; III. Minimizing Costs -
Negative Dynamic Programming; IV. Maximizing Rewards -
Positive Dynamic Programming; V. Average Reward Criterion; VI.
Stochastic Scheduling; and VII. Bandit Processes. The author
presents these subjects in a concise and elegant way, evading
:e hni~ajl;n i and appealing to the imagination of the reader.
The author does not build up a huge overall theory but presents
numerous important examples illustrating the methods and
underlying structures, counterexamples for expected results, and
useful exercises for the reader. The emphasis lies on qualitative and
structural results concerning the optimal policy and the value
function. However, computational approaches are also dealt with,
and the author states that few mathematical prerequisites are needed.
This is true as long as one is willing to accept some intuitively
obvious relations. Without mathematical proofs, it takes some time
continued on following pages

1 111~- I- -- --

~ 111 ----~--- ~----




to r.jll,. understand the foundations. The optimality equation is
proved in a rough way and only for special cases. Furthermore, it
seems possible to prove it for all cases without the heavy measure-
theoretic apparatus using the fact that the action space is finite or
the ..r. 1-.il.,I i, measures have a countable support.
The book is strongly recommended to economists and
operations researchers as well as to mathematicians ready to invest
some additional work.
-M. Schil

Introduction to Sensitivity and Stability Analysis in
Nonlinear Programming
By A.V. Fiacco
Academic Press, New York, 1983
ISBN 0-12-254450-1

This book is concerned with sensitivity and stability issues
related to the following general parametric nonlinear programming
problem .(e: minimize {f(x,E): gi (x,e)> 0 for i=l, ..., m, hj (x,e)
= 0 for j=l, ..., p} where xeEn and where e is a parametric vector in
Ek. Given an e in Ek, a local solution x(e) along with a
corresponding set of Lagrange :iulrplik rs u(E) and w(e) associated
with the inequality and the equality constraints in P(e),
S". 'rti.il,, satisfying the first-order optimality conditions, are
referred to as the Karush-Kuhn-Tucker (KKT) triple, and the function
f*(e) defined as f[x(e), E] is referred to as an optimal value
function. The Scliii, information generated is concerned with
the first-order variations in the KKT triple and with the first-order
and second-order variations in f*(E) with respect to the problem
parameter vector e. This is hence a local perturbation analysis.
The t.ili) analysis, on the other hand, is a finite perturbation
analysis and is concerned with generating parametric bound
information on the optimal value function f*(s) or on the solution
point x(e). The book conducts this type of analysis using both an
algorithm-independent as well as an algorithm-dependent approach
and provides specializations for particular generic cases of P(e) such
as problems involving right-hand-side perturbations only as well as
for particular practical applications.
Following a brief motivational introduction in Chapter 1,
Chapter 2 presents a collection of basic sensitivity and stability
results for P(e) and its various realizations which have been studied
in the literature. These results deal with: (a) the continuity of f*(e)
and of the point-to-set map which maps E to the set of alternative
optimal solutions; (b) differentiability properties of f*(E) based on
a rate of change of this function with respect to e at a solution
point and the existence and computation of directional derivatives
of this optimal value function; (c) the behavior of the KKT triple
with respect to continuity and differentiability properties, and (d)
bounds on optimal value functions and on local solutions to the
parametric problem P(e).
Chapter 3 begins to extend the existing results for generating
the desired sensitivity information. Using standard second-order
,llit.ijcn optimality conditions for a strict local minimum along
with appropriate constraint qualikta.in,, the existence and

behavior of first-order variations of the KKT triple with respect to
the problem parameters are explored, and explicit representations
are developed for these partial derivatives. Based on this, formulae
are derived for first and second-order changes in the optimal value
function f*(e) with respect to the problem parameters e.
The actual numerical aspects related to the efficient
determination of these partial derivatives are addressed in Chapter 4.
The utility of obtaining these derivatives lies in being able to
provide a first-order estimation or representation of the KKT triple,
and a second-order representation of the optimal value function.
This in turn is useful in characterizing the convexity of f*(E) and
for characterizing the stability of a solution to the problem subject
to perturbation.
Chapter 5 continues the analysis of Chapter 4, examining a
special case of a right-hand-side perturbation. It is noted that by
treating E as a variable in P(E), and by adding a constraint e=a, the
foregoing analysis may be viewed as being also related to a
perturbation of the right-hand-side a. Nonetheless, an explicit
treatment of this is provided for the problem P2(E): minimize {f(x):
g(x) >el, h(x) = e2}, where e = (El,E2). A historical digression
first points out the well-known relationships between the variation
of f*(e) with respect to e and the Lagrange multipliers u(e) and
w(E). Thereafter, as before, explicit formulae are developed for first-
order derivatives of the KKT triple and for the first and second-order
derivatives of the optimal value function under stated second-order
assumptions. The utility of these expressions in generating
stability information, as well as its role in cyclical decomposition
schemes for solving nonlinear programming problems is also
discussed. In particular, the use of the second-order results in
designing second-order procedures for generating search directions
in resource-directed decomposition procedures for suitable nonlinear
programming problems is pointed out as an important application
of these results.
In contrast with the algorithm-independent approach of Chapters
3-5, Chapters 6 and 7 demonstrate how standard nonlinear
programming algorithms have an inherent capability of generating
sensitivity information during the normal course of their solution
procedure, provided they terminate with a KKT triple satisfying the
usual first-order optimality conditions. Chapter 6 makes this point
using twice-differentiable penalty functions and, in particular,
employs a logarithmic barrier term with respect to the inequality
constraints, and a quadratic penalty term with respect to the equality
constraints for the purpose of giving a specific illustration. Again,
first-order sensitivity results for the KKT triple are obtained.
Further, directly from the estimates of the gradient and hessian of
the penalty function in the limit as the penalty parameter vanishes,
first and second-order results for the optimal value function are also
obtained. Specializations to right-hand-side perturbations are also
pointed out.
Chapter 7 continues to illustrate this fact using other popular
algorithms including Newton-based procedures, projected-gradient
and reduced-gradient algorithms, and the augmented Lagrangian
methods. The basic idea here is that these and other algorithms
typically determine a solution to a nonlinear program via a sequence
of problems, each of which is some perturbation of the original

---- --- -- --


problem, and hence the solution procedure naturally contains
cl.itiL.iit, information. Besides, having successfully obtained a
KKT triple satisfying first-order conditions, one directly has the
information for the first order derivatives of f*(e) and the KKT
triple, from which second-order derivative information for f*(E) may
be derived. This is done via an estimate of the inverse of the
Jacobian of the first-order necessary optimality conditions, which is
I.r'i-kll. available when employing a Newton-type procedure for
solving the sequence of subproblems in determining a KKT triple.
Chapter 8 employs the penalty function of Chapter 6 in a
computer program SENSUMT, which also performs the associated
sensitivity analysis in order to study a particular application
relating to a multi-item continuous-review inventory system. Brief
mention is also made of the study of other applications including a
geometric programming model of a stream water pollution
abatement system and a nonlinear structural design problem.
Chapter 9 deals with the stability analysis, developing
piecewise linear, continuous, global upper and lower bounding
functions which envelope f*(E), as a function of this parameter E,
when f*(e) is known to be convex or concave. Connections
between the construction mechanism of these optimal value bounds
and duality theory are also explored. A similar scheme can be used
for nonconvex parametric programs whenever suitable convex or
concave underestimating or overestimating problems can be
constructed. This is extended for constructing parametric bounds for
the solution point x(e) as well. Again, it is emphasized that most
nonlinear pr.'grai'n. ng algorithms have an in-built capability of
generating these parametric bounds via the information generated
during the normal course of their operation.
Finally, Chapter 10 looks to future research directions. The
author anticipates movement toward a further unification of theory
which will lead to an extension of the ideas discussed herein to
other areas as well as a transfer of computational techniques from
other developments to analyze sensitivity and stability issues.
The book itself is a major step toward this unification. It is a
first book which deals in a comprehensive fashion with sensitivity
and stability issues related to nonlinear programming problems. It
is based largely on the author's own original work and an extension
of existing analysis which relates to the subject of this book. In
this respect, the book represents a rich collection of ideas and
concepts which are woven together in a clear and well-written
presentation. This book will not only encourage research in this
area, certainly some along the many directions pointed out
throughout the text, but will also encourage the study and
incorporation of sensitivity and stability information generation
capability in standard nonlinear programming algorithmic
developments and codes, and the use and interpretation of this type
of information by practitioners working with various applications.
-H.D. Sherali

Trees and Hills
By R. Greer
North-Holland, Amsterdam, 1984
ISBN 0-444-87578-6

The title of this book is misleading with regard to its contents.
Only the subtitle ("Methodology for Maximizing Functions of
Systems of Linear Relations") informs the reader about the problem
dealt with by the author. In this monograph a set of linear
relations defining a subset of Rd is considered. Any relational
operator {<, <, =, 4, >, >} may define a condition. The problem
treated is to determine all of those vectors x e Rd which satisfy or
do not satisfy elements of this set of linear relations in such
patterns as will extremize certain functions of interest.
This general class of problems includes, for example, the NP-
complete Weighted Closed, Open, or Mixed Hemisphere problem.
Also linear programming falls into this category of problems of
extremizing functions of systems of linear relations.
The present monograph is mainly concerned with the
development of a tree algorithm for solving the general problem.
According to the author, it is the only known nonenumerative
algorithm for solving the problem above, which furthermore
enables the Weighted Open Hemisphere (WOH) problem to be
solved in a much shorter time than by other algorithms. The author
proves this latter assertion by giving some test examples in
Chapter 9.
The monograph extends part of the author's Ph.D. dissertation.
It contains a very detailed tutorial on polyhedral convex cones
(Chapter 2, pp. 15-82!). The tree algorithm is developed in two
stages. First, in Chapter 5, a tree algorithm for solving the WOH
problem is presented. Then after discussing in Chapter 4 how
problems of extremizing functions of systems of linear relations
may be reduced, the general tree algorithm is presented in Chapter
5. In Chapter 6 the computational complexity of the tree alg.'rlliun
is discussed. Another methodology for extremizing functions of
systems of linear relations is compared and contrasted with the tree
algorithm in Chapter 7, and applications of the tree algorithm are
discussed in Chapter 8. A general conclusion is given in the last
This book will be primarily interesting for the experts in the
area mentioned, though a more compact edition would be sufficient
for them. However, due to the very detailed introduction to the area
of polyhedral convex cones and the full description of the WOH
problem, the book will also be useful to the interested reader who is
only slightly familiar with these fields. The uncommon symbols
used in parts do not really make it more difficult to handle.
-H.-J. Kruse



Maintained by the Mathematical Programming Society (MPS)

This Calendar lists noncommercial meet-
ings specializing in mathematical program-
ming or one of its subfields in the general
area of optimization and applications,
whether or not the Society is involved.
(The meetings are not necessarily 'open'.)
Anyone knowing of a meeting that should
be listed here is urged to inform Dr. Philip
Wolfe, IBM Research 33-2, POB 218,
Yorktown Heights, NY 10598, USA;
Telephone 914-945-1642, Telex 137456.
Some of these meetings are sponsored
by the Society as part of its world-wide
support of activity in mathematical
programming.Under certain guidelines
the Society can offer publicity, mailing
lists and labels, and the loan of money to
the organizers of a qualified meeting.
Substantial portions of meetings of other
societies such as SIAM, TIMS, and the
many national OR societies are devoted to
mathematical programming, and their
schedules should be consulted.

September 15-19: International Conference on Stochastic
Programming, Prague, Czechoslovakia.
Contact: Dr. Thomas Cipra, Department of Statistics, Charles
University, Sokolowska 83, 18600 Prague 8, Czechoslovakia.
Cosponsored by the Committee for Stochastic Programming
of the Mathematical Programming Society.

April 6-8: "CO87", a Conference on Combinatorial
Optimization, Southampton, U.K.
Contact Dr. C.N. Potts, Faculty of Mathematical Studies,
University of Southampton, Southampton S09 5NH, United
Kingdom. (Sponsored by the London Mathematical Society.
Deadline for abstracts, 5 January 1987.)

July 6-8: Martin Beale Memorial Symposium, London, U.K.
Contact Professor MJ.D. Powell, Department of Applied
Mathematics and Theoretical Physics, University of Cambridge,
Silver Street, Cambridge (CB3 9EW, United Kingdom.
Telephone (0223) 337889, Telex 81240.

August 29 September 2: Thirteenth International
Symposium on Mathematical Programming in Tokyo, Japan.
Contact Professor Masao Iri (Chairman, Organizing Committee),
Faculty of Engineering, University of Tokyo, Bunkyo-ku, Tokyo
113. Official triennial meeting of the MPS.

20 August 1986

~----~-~ ~

Mail to:
c/o International Statistical Institute
428 Prinses Beatrixlaan
2270 AZ Voorburg, The Netherlands

Dues for 1986 include subscription to Vol. 34-36 of

Dues, payable to MPS, Inc., are Dfl. 125 (or $45, or Sw.Fr. 90,
or FF 340, or DM 112, or 28) for continuing members. For
1986, NEW MEMBERS pay ONE-HALF these rates.

Subscription to
is available to all members for Dfl. 45 (or $15, or Sw.Fr. 33,
or FF 120, or DM 40, or 10).

Q I wish to enroll as a member of the Society. My
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Mailing address (please print)

Student Applications: Dues are one-half the above rates. Have
a faculty member verify your student status below and send
application with dues to the above address.
Faculty verifying status



I ~ _

- -----------pp--~----------LPlsC-sl

I~ --- I

Kenneth O. Kortanek, formerly of Carnegie-Mellon,
has been appointed Murray Research Professor of The
Management Sciences, College of Business Administration,
University of Iowa....Horst Hamacher (Florida) is visiting
the Technical University of Graz, Austria for the academic year
19886-87...Bruce Golden (Maryland) announces the
availability of NETSOLVE, an interactive software package for
network manipulation and optimization.
The October 26-29 ORSA/TIMS meeting in Miami Beach
will feature several 90-minute tutorials on state-of-the-art
mathematical programming topics. Included are lectures
by George Nemhauser (Georgia Tech) on Integer
Programming, Dimitri Bertsekas (MIT) on Network
Algorithms, and Jan Karel Lenstra (Mathematisch Centrum,
Amsterdam) on Sequencing and Scheduling. There will also be
many regular sessions on mathematical programming, including
several on the Karmarkar algorithm. Additionally, Bob
Jeroslow (Georgia Tech) has arranged a set of sessions on the
AI/OR Interface.
Deadline for the next OPTIMA is November 15, 1986.

--- ---- -----1

Books for review should be sent to the
Book Review Editor, Prof. Dr. Achim
Bachem, Mathematiches Institute der
Universitat zu Koln, Weyertal 86-90,
D-5000 K61n, West Germany.

Journal contents are subject to change
by the publisher.

Donald W. Hearn, Editor
Achim Bachem, Associate Editor
Published by the Mathematical
Programming Society and
Publication Services of the
College of Engineering,
University of Florida.
Composition by Lessie McKoy,
Graphics by Lise Drake.

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